As discussed, a key statement for the ABB method is: “if \(\cos (\mathbf {s}_{k-1},\mathbf {y}_{k-1}) \approx 1\) , then \(\mathbf {s}_{k-1}\)  is close to an eigenvector” (see, e.g., [1]}).
The following new result quantifies this statement for the quadratic case.
We need the assumption that \(\lambda \)  is a simple eigenvalue, since otherwise an eigenvector is not uniquely defined.
We consider the situation \(\cos ^2(\mathbf {s}, A\mathbf {s}) \approx 1\) , so that \(\sin (\mathbf {s}, A\mathbf {s})\)  is small.
